1. **Statistical Mechanics**:

Quantum Mechanics:

2. **Ramanujian**:

1 +2 + 3 + …+ n = – 1/12

Note: this formula is used in Quantum Physics dealing with infinity n (although it cancels out each other in subsequent calculations)

**Tau Special Function**:

3. **Boolean Algebra**: George Boole (1847 in 《The Mathematical Analysis of Logic》) used Symbolic variables (not numbers) for Logic, inspired by Galois (1832 in Groups & Finite Fields), Hamilton’s quaternion algebra (1843).

“**AND**”

“**NOT**”

“**XOR**”

“**Extra constraints** ”

4. Solomon Golomb, Sol: “**Linear Feedback Shift Register” (LFSR)** — shift left the first register, fill in the back register with XOR of certain “Taps” (eg.chosen the 1st, 6th, 7th registers)

Maximal Length = The shift register of size n will repeat every steps (exclude all ‘0’ sequence).

Which arrangement of “Taps” will produce the maximal length ?

Solomon applied **Pure Math** : represent the above sequence of registers algebraically by:

in reducible modulo 2 (prime in polynomial, ie can’t be factored).

=> the sequence is Maximal length

**LFSR Applications**: Telecommunications, 3G/4G/5G, CDMA, Wifi, computers, network, signal transmission error-correction CD/DVD, Astrology : Venus-Earth distance, etc.